Calabi-Yau Pairs and Mirror Symmetry for Fano Varieties

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Algebraic geometry has for many decades been one of the core disciplines of mathematics, and the subject remains as vital today as it was 150 years ago as a source of new ideas and important problems. The basic objects studied in algebraic geometry are geometric shapes defined by polynomial equations in an ambient projective space. The Minimal Model Program (MMP) shows that, up to surgery, these shapes can be constructed out of "building blocks" of pure geometric type, and these have:(1) positive curvature (Fano varieties),(2) zero curvature (Calabi-Yau varieties),(3) negative curvature (varieties of general type).These pure geometric types correspond intuitively to the geometry of the sphere, of the Euclidian plane and of the hyperbolic plane. The geometry of each pure type has distinct features and properties: presence of rational curves, behaviour in families, surgery operations between the variety and other varieties.. A very useful technique in recent developments has been to consider perturbations of these pure geometric types. The perturbed geometric types are defined for pairs of a variety and some lower dimensional shape lying on it (for example a slice of the original shape). The varieties of perturbed pure geometric types are called log Fano, log Calabi-Yau, and of log general type. They share many of the features of the associated pure types. The geometry of pairs is very rich: a pair can have a certain perturbed type (log Calabi-Yau for example) while its underlying variety has a different pure geometric type (Fano in our example). The geometry of the pair then blends features of Calabi-Yau and Fano geometries. My research concentrates on varieties and pairs whose geometry or perturbed geometry is of Fano or Calabi-Yau type. These are important in mathematical physics: according to string theory, the fundamental objects in physics are strings rather than point-like particles. These strings move in a background that, in addition to space and time, has extra hidden dimensions curled up in a background variety which is Fano or Calabi-Yau (depending on the version of the theory). Explicitly, this proposal is concerned with the geometry of log Calabi-Yau and Fano shapes. The first project studies transformations of log Calabi-Yau shapes that preserve an additional invariant. The most interesting case is that of transformations between log Calabi-Yau shapes that are made of a Fano shape and a Calabi-Yau slice of it. The transformations are then required to preserve the volume of the Calabi-Yau slice. The second project focusses explicitly on mirror symmetry for Fano varieties. Mirror symmetry is a duality that occurs when two mathematically different background geometries produce the same physics; the two geometries are then called mirror dual. Conjecturally, Fano shapes are mirror symmetric to so called cluster varieties. Cluster varieties are families of log Calabi-Yau shapes that can be glued by the transformations studied in the first project. The proposed research will apply techniques and ideas from algebraic geometry and from the Minimal Model Program to deepen our understanding of cluster varieties and of log Calabi-Yau geometries. In turn, the results of this work will provide a new angle on the geometry of Fano shapes.